Applied Physics and Mathematics Annotation << Back META-ELLIPTICAL ORBITS IN THE CLASSICAL TWO-BODY PROBLEM A.V. KUKUSHKIN Physical and mathematical grounds are presented for a proof of existence of post-Newtonian solutions of the classical two-body problem to generalize the Newton elliptical orbits solution. The Newton approach is being revisited through the kinematics laws of a mass point’s compound motion when the absolute velocity of the body is resulting from the sum of its relative velocity along the elliptical orbit of the eccentricity e and its transfer velocity of its uniform rotation with the angular velocity around the system’s mass center at rest. In the resulting solution is a free parameter of a certain range of permitted values. The range of values is chosen based on the energy characteristics of a compound motion system. These characteristics define the precise boundaries of the domain within the flat { , e} where new solutions exist. Transformations of the orbits found in this domain show that these are meta-elliptical orbits. Orbital motions with the retrograde precession of the apsis line are being demonstrated to violate the introduced energy restrictions and therefore must be dismissed. Keywords: apsidal speed, post-Keplerian equation, eclipsing binary stars, orbital function, rotating complex plane, compound member of sum in the expression of kinetic energy, meta-elliptical orbits, singular sector of the domain, stable external boundary cycle. DOI: 10.25791/pfim.06.2020.1185 Pp. 19-42.